Décomposition en éléments simples

En mathématiques, la décomposition en éléments simples d'une fraction rationnelle (parfois appelée décomposition en fractions partielles) est son expression comme somme d'un polynôme et de fractions J/H où H est un polynôme irréductible et J un polynôme de degré strictement inférieur à celui de H. Cette décomposition est utilisée dans le calcul intégral pour faciliter la recherche des primitives de la fonction rationnelle associée. Elle est aussi utilisée pour calculer des transformées de Laplace inverses. Déterminer quels polynômes sont irréductibles dépend du corps de scalaires utilisé. Ainsi, si les nombres complexes sont utilisés, seuls les polynômes de premier degré seront irréductibles. Si l'on se limite aux nombres réels, les polynômes irréductibles seront de degré 1 ou 2. Si l'on se limite aux nombres rationnels, on pourra trouver des polynômes irréductibles de degré arbitraire ; il en va de même sur les corps finis. Énoncé Sur un corps quelconque U

Low Frequency Sound Field Reconstruction in Non-rectangular Room: A Numerical Validation

Hervé Lissek, Vu Thach Pham, Etienne Thierry Jean-Luc Rivet

Characterization of sound fields in rooms has always been a challenging task. A faithful reconstruction of sound fields in rooms generally would require an impractically high number of microphones. At low frequencies, where sound field can be modeled as a finite superposition of modes, a combination of Matching Pursuit (MP) and Least-squares optimization can help interpolate the Room Impulse Responses using a relatively small number of measurements. Our research focuses on using this paradigm to obtain the low-frequency information of a room in a broader sense, where the spatial distribution of sound pressure in a non-rectangular reverberating room can be reconstructed and visually analyzed. Several evaluations are performed along the progression of the framework to confirm its validity. The modal parameters estimated by MP are also compared with those obtained by the Rational Fraction Polynomial method (global curve fitting). Finally, the reconstruction of sound field, which leads to the visualization of the spatial distribution of sound pressure at any chosen frequency in the range of study, is validated visually and numerically using a finite element method software for a non-rectangular room model. This demonstration provides an extensive look at the high robustness and reliability of the framework as a whole, which is crucial in terms of its practical implementations for room acoustics practitioners.

2018

Blind Audio-Visual Source Separation Using Sparse Redundant Representations

Anna Llagostera Casanovas, Gianluca Monaci, Pierre Vandergheynst

This report presents a new method to confront the Blind Audio Source Separation (BASS) problem, by means of audio and visual information. In a given mixture, we are able to locate the video sources first and, posteriorly, recover each source signal, only with one microphone and the associated video. The proposed model is based on the Matching Pursuit (MP) [18] decomposition of both audio and video signals into meaningful structures. Frequency components are extracted from the soundtrack, with the consequent information about energy content in the time-frequency plane of a sound. Moreover, the MP decomposition of the audio is robust in front of noise, because of its plain characteristic in this plane. Concerning the video, the temporal displacement of geometric features means movement in the image. If temporally close to an audio event, this feature points out the video structure which has generated this sound. The method we present links audio and visual structures (atoms) according to their temporal proximity, building audiovisual relationships. Video sources are identified and located in the image exploiting these connections, using a clustering algorithm that rewards video features most frequently related to audio in the whole sequence. The goal of BASS is also achieved considering the audiovisual relationships. First, the video structures close to a source are classified as belonging to it. Then, our method assigns the audio atoms according to the source of the video features related. At this point, the separation performed with the audio reconstruction is still limited, with problems when sources are active exactly at the same time. This procedure allows us to discover temporal periods of activity of each source. However, with a temporal analysis alone it is not possible to separate audio features of different sources precisely synchronous. The goal, now, is to learn the sources frequency behavior when only each one of them is active to predict those moments when they overlap. Applying a simple frequency association, results improve considerably with separated soundtracks of a better audible quality. In this report, we will analyze in depth all the steps of the proposed approach, remarking the motivation of each one of them.

2006

Model order reduction based on functional rational approximants for parametric PDEs with meromorphic structure

Davide Pradovera

Many engineering fields rely on frequency-domain dynamical systems for the mathematical modeling of physical (electrical/mechanical/etc.) structures. With the growing need for more accurate and reliable results, the computational burden incurred by frequency sweeps has increased too: in many practical cases, a direct frequency-response analysis over a wide range of frequencies is prohibitively expensive. In this respect, model order reduction (MOR) methods are very appealing, as they allow to replace the costly solves of the original problem with a cheap-to-evaluate surrogate model.In this work, we describe a MOR approach, dubbed "minimal rational interpolation" (MRI), that builds a rational interpolant of the frequency response of the dynamical system. In MRI, we build a surrogate model in a data-driven fashion, starting from only few (very expensive) solves of the original problem at well-chosen frequencies. Notably, we do not need any knowledge of (nor access to) the underlying structure of the original problem, so that MRI can be described as a "non-intrusive" method. We perform a theoretical analysis of MRI, showing that it converges to the exact frequency response in a quasi-optimal way, in an "approximation theory" sense. We also describe how this approach can be complemented by adaptive sampling strategies, which, relying on a posteriori error estimators, allow to select automatically the "best" sampling frequencies.Oftentimes, the underlying problem does not depend on frequency alone, but also on additional parameters, which might represent uncertain features of the physical system or design parameters that have to be optimized. This is the so-called "parametric" case, which is much more complex than the non-parametric one, especially if a modest number of parameters is involved. As a way to tackle the parametric setting, we propose a MOR approach based on marginalization: we use MRI to build local frequency surrogates at different parameter configurations, and then we combine these local surrogates to obtain a global reduced model. Several issues arise when carrying our this "combination" step. In this thesis, we propose a practical algorithm for this, relying on matching the partial fraction expansions of the local surrogates term-by-term.Several numerical experiments are carried out as a way to showcase the effectiveness of our proposed approaches, both in the non-parametric and parametric settings. Our "case studies" are selected as simplified versions of problems of practical interest. Notably, we include examples of resonant behavior of mechanical structures with uncertain material properties, and of impedance modeling of distributed electrical circuits with a modest number of design parameters.